Note: Section 15.8 of the text book tells you the ground state energy of this particular isolated potential well. You can also use the band structure software to check the energy. This will save some time when determining the ground state wavefunction (see Chapter 5).

Note: This is a tricky problem to conceptualize. How do I think about a “free particle” that is subject to boundary conditions? Imagine an electron traveling around a circular particle accelerator. The electron comes back to where it started after traveling a distance L. If the electron is in an energy eigenstate, the probability of finding the electron at a certain location is equally distributed around the circle. The eigenstate wavefunction must be single-valued (only one value for each position), therefore, the wave vector that is part of the eigenstate wavefunction is restricted to certain values.